reserve x,y for set,
        r,s for Real,
        n for Nat,
        V for RealLinearSpace,
        v,u,w,p for VECTOR of V,
        A,B for Subset of V,
        Af for finite Subset of V,
        I for affinely-independent Subset of V,
        If for finite affinely-independent Subset of V,
        F for Subset-Family of V,
        L1,L2 for Linear_Combination of V;

theorem Th18:
  v in If implies ((center_of_mass V).If |-- If).v = 1/card If
  proof
    consider L be Linear_Combination of If such that
    A1: Sum L=1/card If*Sum If & sum L=1/card If*card If and
    A2: L=(ZeroLC V)+*(If-->1/card If) by Th15;
    assume A3: v in If;
    then A4: conv If c=Affin If & (center_of_mass V).If in conv If
      by Th16,RLAFFIN1:65;
    (center_of_mass V).If=Sum L & sum L=1 by A1,A3,Def2,XCMPLX_1:87;
    then dom(If-->1/card If)=If & L=(center_of_mass V).If|--If
      by A4,RLAFFIN1:def 7;
    hence ((center_of_mass V).If|--If).v=(If-->1/card If).v by A2,A3,FUNCT_4:13
    .=1/card If by A3,FUNCOP_1:7;
  end;
